Acta of Bioengineering and Biomechanics Vol. 8, No. 1, 2006

Is a “movable hinge axis” used by the human stomatognathic system? K. M. THIEME, D. KUBEIN-MEESENBURG, D. IHLOW, H. NÄGERL Department of Orthodontics, Georg-August-University of Göttingen, Robert-Koch-Str. 40, D-37099 Göttingen, Germany. Tel.: +49-551 – 39-9697, +49-551 – 39-8344. Fax: +49-551 – 39-8350. E-mail: [email protected]; [email protected]; [email protected]; [email protected] This treatise deals with sagittal in vivo motions of the human mandible. The concept of a “movable hinge axis”, which is commonly used in dentistry, was scrutinised theoretically and empirically. We wondered whether a “movable hinge axis”– or better a mandibularly fixed hinge axis (MFHA) – was actually used by subjects with sound temporomandibular joints. To answer this question we first showed that the assumptions of a MFHA would comprise that of the neuromuscular apparatus of the stomatognathic system piloting the mandible by solely two kinematical degrees of freedom (DOF). We spatially recorded in vivo motions of mandibles with high-precision ultrasonic devices. The subjects were asked to guide their mandibles in sagittal movements so that the lower incisal edges ran along the Posselt diagrams. The mathematical procedure is described in detail, hence a possible use of two DOF by a subject could quickly be puzzled out from a set of motions. These analyses revealed that the quasi-plane mandibular movements were approximately piloted by two kinematical DOF in subjects with sound temporomandibular joints. The grade of approximation was measured. Thus, the ensemble of possible positions of the moved body (mandible) can be described by a coordinate system, which is inherent in the stomatognathic system. Lacking precision and poor reproducibility in using only two variables for mandibular position control yield hints that the subject has clinical problems in his stomatognathic system. Key words: human mandible, mandibular movements, movable hinge axis, degrees of freedom, inherent coordinate system, neuromuscular system

1. Introduction Up to now axiography is a major part of instrumental analyses in clinical dental practices to evaluate functional states of the stomatognathic system. It is said to

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record the path of the so-called “movable hinge axis” of the mandible. The procedure is as follows: The dental surgeons mostly guide the patient’s mandible out of centric occlusion (CO) in a small movement parallel to the sagittal plane and thus produce a finite rotational axis in the region of the temporomandibular joint (TMJ). If its lateral projection looks like a point this axis is said to be the “movable hinge axis“ which would remain stationary in the mandible and abount which the mandible would rotate [1]–[4]. This statement was often criticized [5]–[9]. NÄGERL et al. [10] have proved that this axiographically defined “movable hinge axis” physically makes no sense since it has no prominent kinematical significance compared with other lines connecting two mandibular points. To find definitely a mandibularly fixed axis, if such exists, the following approach has to be adopted: Firstly the subjects should be able to perform quasi-plane sagittal mandibular movements keeping the three degrees of freedom (DOF) negligibly small belonging to lateral shift, horizontal and frontal rotation. The ensemble of possible positions of the moved body (mandible) in the reference system (maxilla) is then given by the position of an arbitrarily taken mandibular point (2 DOF), which lies within a plane domain whose margin consists of a closed curve and the mandibular rotation (1 DOF). But, if a distinct mandibular point exists, whose domain is degenerated to a pure line segment, the criterion for the existence of the “movable hinge axis” – or better mandibularly fixed hinge axis (MFHA) – would be fulfilled. Only then the subject would reduce the control of mandibular positions to 2 DOF: The position of the point

Fig. 1. Motion paths of selected mandibular points of patient KS, projected in the sagittal-vertical plane seen from the right. Point PLI represented the lower incisor edge which followed the Posselt motion. Point PC at (0, 0) is located near the condyle’s centre on the right side of patient’s head. Nearby this point the mandibular points ran around very small areas.

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Point Pmin moved on the same path there and back while opening and closing the mouth. Remark the change of the sense of circulation for points above and below Pmin

on this line segment (arc length) and the rotation angle. For any cyclic movement of the mandible this special point must move on this line segment back and forth. We would like to demonstrate by in vivo measurements whether or not the reduction from 3 to 2 DOF of the mandible’s position may actually be possible. An 11-year old girl was asked to pilot her mandible as much as possible along the borders of the domain of the mandibular positions parallel to the sagittal plane: The starting position was CO. She moved her mandible to the most anterior possible position under teeth contact, opened her mouth as far as possible, and closed her mouth in a posterior motion to reach CO again. She complied with our requests and piloted her mandible along far distant positions in a plane fashion as we could check by the spatial kinematical measurements. The paths of some mandibular points were calculated from the recorded data (figure 1). The edge of the lower incisor PLI ran around the area of the well-known Posselt diagram [11]. The path of the point PC (located near the condyle’s centre) enclosed a domain just as the paths of the points P3, P4, P5, P6. For the point Pmin, however, the domain seemed to be degenerated to a line segment on which this point ran there and back. At first sight Pmin fulfilled the criterion of a maxillarily movable and mandibularly fixed hinge axis as described above. The girl was apparently able to pilot diverse plane movements between mandibular positions having long distances from each other by using only 2 DOF: A position i of the mandible can be described by the pair (Li, α i). Li is the arc length covered by the hinge axis Pmin along its path starting from CO (L0 = 0, α 0 = 0°), and α i is the angle of mouth opening in relation to CO (figure 2).

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K. M. THIEME et al. Fig. 2. The motion path of point Pmin could closely be approximated by a circle with the centre C and radius R. L is the arc length covered by Pmin, α is the angle of mouth opening in relation to centric occlusion

In the following we describe the mathematical procedure by which the possible reduction from 3 to 2 DOF of the mandibular movements can be figured out from spatial in vivo measurements and by which the grade of approximation of using a MFHA by the neuromuscular system can be estimated.

2. Material and mathematical method 2.1. Experimental apparatus To record spatial motions of mandibles in vivo, we first used the ultrasonic device MT1602 (Dr. Hansen & Co. Bonn, Germany [12]) and later on a more precise CMSJMA (Zebris Medizintechnik, Isny, Germany) (figure 3). Both measurement systems are able to record the spatial movement of the mandible in relation to the maxilla with 6 DOF so that the spatial path of each mandibular point can be calculated.

Fig. 3. The ultrasonic measurement system CMS-JMA in a young patient in-situ: The transmitter antenna with three sensors was fixed to the lower incisors, while the receiver antenna with four sensors was attached above the nose to the head

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2.2. The subjects Up to now we have measured more than 120 subjects with the MT1602 and more than 30 subjects with the CMS-JMA as described by way of introduction. To figure out the number of DOF used for mandible control we scrutinised in detail quasi-plane mandibular movements of 17 adult persons without orthodontic treatment classified as class I and 20 young and adult class-I-patients after orthodontic treatment recorded by the MT1602 as well as 28 young class-II-patients before orthodontic treatment measured by the CMS-JMA. 2.3. Search for the mandibularly fixed hinge axis (MFHA) We took into account the shapes and values of the areas around which mandibular points drove (figure 1): The lower incisal edge PLI drove clockwise along its Posselt diagram just like other mandibular points nearby (P1) yielding mathematically negative areas A2. The points in the posterior region (P2), however, drove anticlockwise yielding A2 > 0. The points PC, P3, P4, P5, P6 ran along loops. Therefore A2 j + A 2k . the areas were composed of positive ( j) and negative (k) parts: A 2 = The loops of the points P3 and P4, above and below PC, showed opposite sense of circulation. These observations made on all subjects suggested the following qualitative statements: 1. With regard to the sense of circulation a line l0 must exist which separates the positive from the negative areas A2. This line l0 is the geometric locus of the mandibular points which run along loops whose positive and negative partial areas A2 j + A 2 k = 0 with A2j > 0 and A2k < 0. These observations add up to zero: A 2 = correspond to the theorem of Steiner (1840) of plane kinematics: The geometric locus of the points of the moved plane (mandible), whose closed paths surround areas A2 of the same size, forms a circle. The circles of different sizes of A2 have the common centre S, the so-called Steiner point [13]. A2 j + | A2 k | we arrive at the 2. Considering the absolute areas A1 = conclusion that among the points of the line l0 there must exist a point Pmin(l0), whose path encloses a minimal absolute area A1min, since in comparison with the cranial points of the line l0 its caudal points ran along their loops with A2 = 0 in opposite sense of circulation. 3. If the absolute area A1min of the point Pmin was found to be zero (A1min = 0) the subject has only used 2 DOF for piloting the mandible and actually adjusted a MFHA.

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The calculation of the area A2 needs less computer programming and run time than the calculation of the absolute area A1 since for this calculation the crossing points of the loops have to be determined additionally. To find the point Pmin(A1min) the following three steps kept the computational time short. Step 1: A point within the domain of the condyle (PC) served as the centre for a square of 10 10 cm2 in the sagittal-vertical plane (x, y). This square was subdivided by a square net with a step width of 0.5 cm. By means of the measured data the path of every net point P(x, y) was calculated. According to the clock frequency of measurement the path shaped up as a series of n points defining a polygon with n vertices CP(xi, yi). The mathematical area of this polygon was split up in triangles which added up to the area A2: A2 =

1 ⋅ 2

i =n−2 i =0

( xi ⋅ yi +1 − xi +1 ⋅ yi ) + xn −1 ⋅ y0 − x0 ⋅ yn −1 .

The calculated function A2(x, y) of the net points P(x, y) was then approximated by means of the method of least squares to the plane A2Plane(x, y). A2Plane(x, y) = 0 yielded the straight line sl0. We settled upon this procedure because a straight line sl0 was commonly found to be a good approximation of the circular segment line l0 between negative and positive areas A2. Step 2: On the straight line sl0 we searched for the point Pmin(sl0) whose path enclosed the minimal absolute area A1. For this purpose we fitted a parabola to the function A1(P(sl0)) using Brent’s method [14]. According to the golden section search the procedure jumped back and forth and found very fast the minimum value of A1(P(sl0)). This special procedure was considered to be very favourable because the searched point Pmin was commonly found to be closely positioned to point Pmin(sl0) (see below). Step 3: In order to find finally the point Pmin, we searched in the neighbourhood of the point Pmin(sl0) using Powell’s method [14] in two preferred directions: Vertical to the straight line sl0 and parallel to it. This procedure was a combination of multidimensional and one-dimensional minimization. Mostly it was sufficient to calculate the minimal value of the area A1 once vertical to the straight line sl0 and once again parallel. Rarely a second iteration was necessary.

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3. Results 3.1. The straight line sl0 In step 1 of our procedure, the function A2(x, y) was favourably fitted by the plane A2Plane(x, y) since A2(x, y) mostly represented nearly a plane. Figure 4 demonstrates this fact by the girl’s data (figure 1). The functions A2(x, y = +5 cm, 0, and –5 cm) nearly formed straight lines. Consequently the contour lines of A2(x, y) = constant nearly formed straight lines, too, which were parallel to the straight line sl0(A2 = 0) (figure 5).

Fig. 4. Data of patient KS: Areas A2 calculated from the paths of the net points P(x, y = const.) with y ∈ (+5 cm, 0, –5 cm). The graphs A2(x, y = const.) almost formed straight lines

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Fig. 5. Data of patient KS: A square of 10 10 cm2 was laid around the point PC. For net points of this square (distance = 0.1 cm) the paths and the corresponding areas A2 were calculated. In excellent approximation, the contour line graph (upper part) yielded straight lines including straight line sl0 (for area A2 = 0). A2(x, y) practically represented a plane (lower part)

For most of the 37 young and adult class-I-patients and the 28 young class-IIpatients the contour lines for A2 in the region of 10 10 cm2 were straight lines in the demonstrated good approximation. Hence, the approximation of the parting line l0 for the positive and negative areas A2 by the straight line sl0 was found to be the useful step 1 in our mathematical procedure. 3.2. The area A1min of the point Pmin The girl’s data yielded the absolute areas A1(sl0) for the points of the straight line sl0 (figure 6). Two minima were seen. The main minimum belonged to the point Pmin in figure 1; the second corresponded to the point P5.

Fig. 6. Data of patient KS: The absolute areas A1 were calculated for the paths of those points which lay on the straight line sl0. The curve showed two minima. The main minimum corresponded to the point Pmin, the second minimum belonged to the point P5 in figure 1

The absolute areas A1(x, y) were plotted for the net points in the 10 10 cm2 square (figure 7) as was already done in figure 5 for A2(x, y). The A1(x, y)-diagrams revealed a valley where the deepest point was the searched minimum. In the contour line graph, the straight line sl0 is added which was often found to run through the two minima as demonstrated in figure 6.

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Fig. 7. Data of patient KS: A square of 10 10 cm2 was laid around the point PC. For net points of this square (distance = 0.1 cm) the paths and the corresponding areas A1 were calculated. In the contour line graph (upper plot), a main minimum and a second minimum were found through which the straight line sl0 ran. The three-dimensional graph correspondingly showed two hollows in the valley (lower part)

Investigating the kinematical properties of the points Pmin and P5 in more detail we found that: 1. In all subjects with sound TMJs, Pmin moved exclusively forth in the course of mouth opening and exclusively back in the course of mouth closing. 2. Pmin normally ran apparently along the same path there and back. 3. The path of Pmin normally showed a circle-like curvature. These findings apparently checked with the expected properties of a MFHA. P5 (the second minimum) did not show these features: 1. The path was not at all circle-like. 2. The point already ran back during anterior mouth opening (figure 1). P5 was always caudal of Pmin and mostly found within the 10 10 cm2 square. In almost all cases, A1(P5) was larger than A1(Pmin). Since we had to reckon with two minima, we had to prevent the mathematical procedure of step 2 from taking the second as the searched minimum of A1(x, y). Brent’s method therefore started at first at the lower end of the straight line sl0 and then again at the upper end. The minimum positioned more cranially was always taken for the further evaluation. In step 3, we searched the absolute minimum of A1(x, y) in the region of the point Pmin(sl0) based on the findings that there the lines A1(x, y) = constant were almost parallel and therefore grad(A1(x, y)) was almost vertical to the straight line sl0 resulting from A2(x, y) = 0. Table. The statistical values for (a) the distance dmin between the point Pmin(sl0) and the point Pmin(A1min) and (b) the distance dref between the point PC and the point Pmin(A1min)

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Device

K. M. THIEME et al. (a) Distance dmin between Pmin(sl0) and Pmin(A1min)

(b) Distance dref between PC and Pmin(A1min)

MT1602

Median cm 0.04

Mean cm 0.07

Std. dev. cm 0.07

Min cm 0.00

Max cm 0.36

Median cm 1.32

Mean cm 1.53

Std. dev. cm 0.81

Min cm 0.32

Max cm 3.66

CMS-JMA

0.02

0.04

0.05 F = 1.96 p < 5%

0.00

0.29

0.61

0.56

0.26 F = 9.71 p < 5%

0.04

1.11

The distance dmin between Pmin(A1min) and Pmin(sl0) was found to be small in each case (table), while the distance d ref between Pmin(A1min) and PC was large. Hence Pmin(A1min) was not located in the condyle’s centre. 3.3. Is the area A1min(Pmin) = 0? This question should be answered with yes if Pmin(A1min) was a point of an actually existing MFHA. Serious problems arose: On the one hand, the area A1min(Pmin) was unavoidably larger than zero because of measuring errors. On the other hand, we regarded the MFHA primarily not given by anatomical mechanical constraints but produced by piloting the mandible by the neuromuscular system. Because of this the condyles can be withdrawn a little bit from the os temporale giving thus the TMJ a certain articulating space [15]. Therefore we expected that the neuromuscular system was only able to implement a MFHA with certain uncertainty. To handle this problem we estimated the very upper limit of the error for the area A1min(Pmin)n of each person n and compared the distributions of these limits with that of the devices. The limits were determined in the following way: 1. We took the confidence interval (CI ) into account for determining a point: CI was 0.01 cm for the MT1602 [12] and 0.003 cm for the CMS-JMA [16]. 2. We calculated the individual path length Lpn by adding up the distances between the vertices CP(xi, yi) of the polygon (used for the calculation of A1min(Pmin)n) on the way there and back: L pn =

i =n−2 i =0

( xi +1 − xi ) 2 + ( yi +1 − yi ) 2 .

3. The rectangle Alimit n = CI · Lpn/2 was regarded as the individual upper limit of error. 4. The difference Dn = A1min(Pmin)n – Alimit n = A1min(Pmin)n – (CI · Lpn/2) was used as the test parameter to check whether the area A1min(Pmin) exceeded its upper limit of error.

Is a “movable hinge axis” used by the human stomatognathic system? a)

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b)

Fig. 8. Histograms of the A1min-distributions: (a) of the 37 young and adult class-I-patients measured with the MT1602 and of the 4 movement cycles of the young class-I-patient HK and (b) of the 28 young class-II-patients measured with the CMS-JMA and of the 8 movement cycles of the young class-II-patient KS. The upper limit of error was smaller than the median for the ensemble as well as for the individual

For both samples (MT1602 and CMS-JMA measurements) the median of the A1min(Pmin)-distribution lay above the respective very upper limit of error (figure 8a, b). In particular, for the more precise CMS-JMA measurements all individual A1min(Pmin)n lay above the upper limit of error. The Student-test of the variable Dn yielded: t = 8.25 (t(5%) = 1.99) for the MT1602 measurements and t = 12.58 (t(5%) = 2.00) for the CMS-JMA. Hence, the area A1min(Pmin) was found not to equal zero! This finding was supported by individual repeated measurements: t = 4.68 (t(5%) = 2.37) for patient HK and t = 8.40 (t(5%) = 2.13) for patient KS.

4. Discussion

All subjects made quasi-plane mandibular motions. The 3 DOF of horizontal and frontal rotation and lateral shift were measurable but small. We wondered whether the stomatognathic system used a further reduction from 3 to 2 DOF in a similar approximation. We could affirm this. The paths of the MFHA in mouth opening and closure almost coincided though the mandible was guided along strongly differing positions. The MT1602 and the CMS-JMA groups showed mean

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residual areas of ~0.048 cm² and ~0.025 cm², respectively, and path lengths of ~3.75 cm and ~2.09 cm. Hence the spacing between back and forth (0.048 cm2 / 3.75 cm = 0.013 cm and 0.025 cm2 / 2.09 cm = 0.012 cm, respectively) corresponded to the breadth of a thin pencil line. Therefore the neuromuscular systems were able to differentiate the entire ensemble of open mouth positions with high precision despite using practically 2 DOF. The mandibular positions could be definitely specified by two variables, the arc length L covered by the MFHA and the rotational angle in relation to the maxilla (figure 2). Thus an orthogonal coordinate system is obtained in which the ensemble of the possible positions of the rigid body is defined one-to-one. This system is independent of the coordinate system of the measuring apparatus and inherent in the stomatognathic system. It makes intra- and interindividual comparisons of mandibular movements possible and easy. Especially the structure of the guidance by the stomatognathic neuromuscular system can be evaluated when the mandible was brought from a starting to a final position [17]. Since according to Steiner the ensemble of the functions A2(x, y) = constant have to represent concentric circles and since we found in the most cases that A2(x, y) = 0 could be replaced by a straight line (sl0), the Steiner centre lay far away from the mandible. In the few cases having a nearer Steiner centre, the search for Pmin(A1min) required additional computational steps. The existence of A2(x, y) = 0 is common to plane kinematics with 3 DOF. It does not imply that a mandibular point Pmin must exist whose absolute area A1(Pmin) equals zero. A1(Pmin) = 0 is the criterion of plane movements with 2 DOF. The residual A1(Pmin) characterizes the approximation of the measured movements to plane movements with 2 DOF. It was found to be close. The data of the few patients with lacking precision and poor reproducibility in using the MFHA and having high residuals A1min(Pmin) gave hints that these patients have hidden problems with their neuromuscular apparatus. In this regard our method to evaluate in vivo the patients’ MFHA yields a novel diagnostic tool of clinical value by determining the area A1min(Pmin).

5. Conclusions Though an ideal MFHA could not be found, we could show that the stomatognathic system normally adjusts and uses a MFHA with a surprisingly high precision to guide the mandible’s position in the case of mouth opening and closing by using only 2 main DOF. Thus an inherent coordinate system of mandibular movements can be determined. The mathematical procedure for finding out the inherent coordinate system can be used for quasi-plane movements of other joint systems by analogy.

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Acknowledgements The authors wish to thank Professor Rainer Schwestka-Polly who did all the measurements with the MT1602. This work was supported by a grant from the Deutsche Forschungsgemeinschaft (SCHW 427/21, 2-2, and KU 535/6-3, respectively) and is permitted by the ethical commission (8/10/02).

References [1] YUSTIN D. C., RIEGER M. R., MCGUCKIN R. S., CONNELLY M. E., Determination of the existence of hinge movements of the temporomandibular joint during normal opening by Cine-MRI and computational digital addition, Journal of Prosthodontics, 1993, Vol. 2 (3), pp. 190–195. [2] NAEIJE M., HUDDLESTON-SLATER J. J., LOBBEZOO F., Variation in movement traces of the kinematic center of the temporomandibular joint, Journal of Orofacial Pain, 1999, Vol. 13 (2), pp. 121–127. [3] TRAVERS K. H., BUSCHANG P. H., HAYASAKI H., THROCKMORTON G. S., Associations between incisor and mandibular condylar movements during maximum mouth opening in humans, Archives of Oral Biology, 2000, Vol. 45 (4), pp. 267–275. [4] NAGY W. W., SMITHY T. J., WIRTH C. G., Accuracy of a predetermined transverse horizontal mandibular axis point, Journal of Prosthetic Dentistry, 2002, Vol. 87 (4), pp. 387–394. [5] LE PERA F., Determination of the hinge axis, Journal of Prosthetic Dentistry, 1967, Vol. 14, pp. 651– 666. [6] BOSMAN A. E., Hinge axis determination of the mandible, PhD Thesis, Utrecht, The Netherlands, 1974. [7] HELLSING G., HELLSING E., ELIASSON S., The hinge axis concept: a radiographic study of its relevance, Journal of Prosthetic Dentistry, 1995, Vol. 73 (1), pp. 60–64. [8] FERRARIO V. F., SFORZA C., MIANI A. Jr., SERRAO G., TARTAGLIA G., Open–close movements in the human temporomandibular joint: does a pure rotation around the intercondylar hinge axis exist? Journal of Oral Rehabilitation, 1996, Vol. 23 (6), pp. 401–408. [9] MORNEBURG T., PRÖSCHEL P. A., Differences between traces of adjacent condylar points and their impact on clinical evaluation of condyle motion, International Journal of Prosthodontics, 1998, Vol. 11 (4), pp. 317–324. [1] NÄGERL H., KUBEIN-MEESENBURG D., SCHWESTKA-POLLY R., THIEME K. M., FANGHÄNEL J., MIEHE B., Functional condition of the mandible: physical structures of free mandibular movement, Annals of Anatomy, 1999, Vol. 181, pp. 41–44. [2] POSSELT U., Studies in the mobility of the human mandible, Acta Odontology Scandinavia, 1952, Vol. 10 (10), pp. 13–150. [3] KLAMT B., NÄGERL H., KUBEIN-MEESENBURG D., Vergleichende Untersuchung von Meßmethoden zur Aufzeichnung der räumlichen Mandibulabewegung (summary in English), Deutsche Zahnärztliche Zeitung, 1990, Vol. 45, pp. 33–35. [4] BLASCHKE W., MÜLLER H. R., Ebene Kinematik, V. R. Oldenbourg, München, 1956, p. 115. [5] PRESS W. H., FLANNERY B. P., TEUKOLSKY S. J., VETTERLING W. T., Numerical Recipes in C – The Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1988, pp. 290–353. [6] KUBEIN-MEESENBURG D., NÄGERL H., KLAMT B., The biomechanical relation between incisal and condylar guidance in man, Journal of Biomechanics, 1988, Vol. 21 (12), pp. 997–1009. [7] HUGGER A., BÖLÖNI E., BERNTIEN U., STÜTTGEN U., Accuracy of an ultrasonic measurement system for a jaw movement recording, Proceedings of the IADR/CED 35th Annual Meeting, Montpellier, 1999.

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[8] NÄGERL H., KUBEIN-MEESENBURG D., FANGHÄNEL J., THIEME K. M., KLAMT B., SCHWESTKA-POLLY R., Elements of a general theory of joints. 6. General kinematical structure of mandibular movements, Anatomischer Anzeiger, 1991, Vol. 173, pp. 249–264.

Is a “movable hinge axis” used by the human stomatognathic system? K. M. THIEME, D. KUBEIN-MEESENBURG, D. IHLOW, H. NÄGERL Department of Orthodontics, Georg-August-University of Göttingen, Robert-Koch-Str. 40, D-37099 Göttingen, Germany. Tel.: +49-551 – 39-9697, +49-551 – 39-8344. Fax: +49-551 – 39-8350. E-mail: [email protected]; [email protected]; [email protected]; [email protected] This treatise deals with sagittal in vivo motions of the human mandible. The concept of a “movable hinge axis”, which is commonly used in dentistry, was scrutinised theoretically and empirically. We wondered whether a “movable hinge axis”– or better a mandibularly fixed hinge axis (MFHA) – was actually used by subjects with sound temporomandibular joints. To answer this question we first showed that the assumptions of a MFHA would comprise that of the neuromuscular apparatus of the stomatognathic system piloting the mandible by solely two kinematical degrees of freedom (DOF). We spatially recorded in vivo motions of mandibles with high-precision ultrasonic devices. The subjects were asked to guide their mandibles in sagittal movements so that the lower incisal edges ran along the Posselt diagrams. The mathematical procedure is described in detail, hence a possible use of two DOF by a subject could quickly be puzzled out from a set of motions. These analyses revealed that the quasi-plane mandibular movements were approximately piloted by two kinematical DOF in subjects with sound temporomandibular joints. The grade of approximation was measured. Thus, the ensemble of possible positions of the moved body (mandible) can be described by a coordinate system, which is inherent in the stomatognathic system. Lacking precision and poor reproducibility in using only two variables for mandibular position control yield hints that the subject has clinical problems in his stomatognathic system. Key words: human mandible, mandibular movements, movable hinge axis, degrees of freedom, inherent coordinate system, neuromuscular system

1. Introduction Up to now axiography is a major part of instrumental analyses in clinical dental practices to evaluate functional states of the stomatognathic system. It is said to

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record the path of the so-called “movable hinge axis” of the mandible. The procedure is as follows: The dental surgeons mostly guide the patient’s mandible out of centric occlusion (CO) in a small movement parallel to the sagittal plane and thus produce a finite rotational axis in the region of the temporomandibular joint (TMJ). If its lateral projection looks like a point this axis is said to be the “movable hinge axis“ which would remain stationary in the mandible and abount which the mandible would rotate [1]–[4]. This statement was often criticized [5]–[9]. NÄGERL et al. [10] have proved that this axiographically defined “movable hinge axis” physically makes no sense since it has no prominent kinematical significance compared with other lines connecting two mandibular points. To find definitely a mandibularly fixed axis, if such exists, the following approach has to be adopted: Firstly the subjects should be able to perform quasi-plane sagittal mandibular movements keeping the three degrees of freedom (DOF) negligibly small belonging to lateral shift, horizontal and frontal rotation. The ensemble of possible positions of the moved body (mandible) in the reference system (maxilla) is then given by the position of an arbitrarily taken mandibular point (2 DOF), which lies within a plane domain whose margin consists of a closed curve and the mandibular rotation (1 DOF). But, if a distinct mandibular point exists, whose domain is degenerated to a pure line segment, the criterion for the existence of the “movable hinge axis” – or better mandibularly fixed hinge axis (MFHA) – would be fulfilled. Only then the subject would reduce the control of mandibular positions to 2 DOF: The position of the point

Fig. 1. Motion paths of selected mandibular points of patient KS, projected in the sagittal-vertical plane seen from the right. Point PLI represented the lower incisor edge which followed the Posselt motion. Point PC at (0, 0) is located near the condyle’s centre on the right side of patient’s head. Nearby this point the mandibular points ran around very small areas.

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Point Pmin moved on the same path there and back while opening and closing the mouth. Remark the change of the sense of circulation for points above and below Pmin

on this line segment (arc length) and the rotation angle. For any cyclic movement of the mandible this special point must move on this line segment back and forth. We would like to demonstrate by in vivo measurements whether or not the reduction from 3 to 2 DOF of the mandible’s position may actually be possible. An 11-year old girl was asked to pilot her mandible as much as possible along the borders of the domain of the mandibular positions parallel to the sagittal plane: The starting position was CO. She moved her mandible to the most anterior possible position under teeth contact, opened her mouth as far as possible, and closed her mouth in a posterior motion to reach CO again. She complied with our requests and piloted her mandible along far distant positions in a plane fashion as we could check by the spatial kinematical measurements. The paths of some mandibular points were calculated from the recorded data (figure 1). The edge of the lower incisor PLI ran around the area of the well-known Posselt diagram [11]. The path of the point PC (located near the condyle’s centre) enclosed a domain just as the paths of the points P3, P4, P5, P6. For the point Pmin, however, the domain seemed to be degenerated to a line segment on which this point ran there and back. At first sight Pmin fulfilled the criterion of a maxillarily movable and mandibularly fixed hinge axis as described above. The girl was apparently able to pilot diverse plane movements between mandibular positions having long distances from each other by using only 2 DOF: A position i of the mandible can be described by the pair (Li, α i). Li is the arc length covered by the hinge axis Pmin along its path starting from CO (L0 = 0, α 0 = 0°), and α i is the angle of mouth opening in relation to CO (figure 2).

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K. M. THIEME et al. Fig. 2. The motion path of point Pmin could closely be approximated by a circle with the centre C and radius R. L is the arc length covered by Pmin, α is the angle of mouth opening in relation to centric occlusion

In the following we describe the mathematical procedure by which the possible reduction from 3 to 2 DOF of the mandibular movements can be figured out from spatial in vivo measurements and by which the grade of approximation of using a MFHA by the neuromuscular system can be estimated.

2. Material and mathematical method 2.1. Experimental apparatus To record spatial motions of mandibles in vivo, we first used the ultrasonic device MT1602 (Dr. Hansen & Co. Bonn, Germany [12]) and later on a more precise CMSJMA (Zebris Medizintechnik, Isny, Germany) (figure 3). Both measurement systems are able to record the spatial movement of the mandible in relation to the maxilla with 6 DOF so that the spatial path of each mandibular point can be calculated.

Fig. 3. The ultrasonic measurement system CMS-JMA in a young patient in-situ: The transmitter antenna with three sensors was fixed to the lower incisors, while the receiver antenna with four sensors was attached above the nose to the head

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2.2. The subjects Up to now we have measured more than 120 subjects with the MT1602 and more than 30 subjects with the CMS-JMA as described by way of introduction. To figure out the number of DOF used for mandible control we scrutinised in detail quasi-plane mandibular movements of 17 adult persons without orthodontic treatment classified as class I and 20 young and adult class-I-patients after orthodontic treatment recorded by the MT1602 as well as 28 young class-II-patients before orthodontic treatment measured by the CMS-JMA. 2.3. Search for the mandibularly fixed hinge axis (MFHA) We took into account the shapes and values of the areas around which mandibular points drove (figure 1): The lower incisal edge PLI drove clockwise along its Posselt diagram just like other mandibular points nearby (P1) yielding mathematically negative areas A2. The points in the posterior region (P2), however, drove anticlockwise yielding A2 > 0. The points PC, P3, P4, P5, P6 ran along loops. Therefore A2 j + A 2k . the areas were composed of positive ( j) and negative (k) parts: A 2 = The loops of the points P3 and P4, above and below PC, showed opposite sense of circulation. These observations made on all subjects suggested the following qualitative statements: 1. With regard to the sense of circulation a line l0 must exist which separates the positive from the negative areas A2. This line l0 is the geometric locus of the mandibular points which run along loops whose positive and negative partial areas A2 j + A 2 k = 0 with A2j > 0 and A2k < 0. These observations add up to zero: A 2 = correspond to the theorem of Steiner (1840) of plane kinematics: The geometric locus of the points of the moved plane (mandible), whose closed paths surround areas A2 of the same size, forms a circle. The circles of different sizes of A2 have the common centre S, the so-called Steiner point [13]. A2 j + | A2 k | we arrive at the 2. Considering the absolute areas A1 = conclusion that among the points of the line l0 there must exist a point Pmin(l0), whose path encloses a minimal absolute area A1min, since in comparison with the cranial points of the line l0 its caudal points ran along their loops with A2 = 0 in opposite sense of circulation. 3. If the absolute area A1min of the point Pmin was found to be zero (A1min = 0) the subject has only used 2 DOF for piloting the mandible and actually adjusted a MFHA.

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The calculation of the area A2 needs less computer programming and run time than the calculation of the absolute area A1 since for this calculation the crossing points of the loops have to be determined additionally. To find the point Pmin(A1min) the following three steps kept the computational time short. Step 1: A point within the domain of the condyle (PC) served as the centre for a square of 10 10 cm2 in the sagittal-vertical plane (x, y). This square was subdivided by a square net with a step width of 0.5 cm. By means of the measured data the path of every net point P(x, y) was calculated. According to the clock frequency of measurement the path shaped up as a series of n points defining a polygon with n vertices CP(xi, yi). The mathematical area of this polygon was split up in triangles which added up to the area A2: A2 =

1 ⋅ 2

i =n−2 i =0

( xi ⋅ yi +1 − xi +1 ⋅ yi ) + xn −1 ⋅ y0 − x0 ⋅ yn −1 .

The calculated function A2(x, y) of the net points P(x, y) was then approximated by means of the method of least squares to the plane A2Plane(x, y). A2Plane(x, y) = 0 yielded the straight line sl0. We settled upon this procedure because a straight line sl0 was commonly found to be a good approximation of the circular segment line l0 between negative and positive areas A2. Step 2: On the straight line sl0 we searched for the point Pmin(sl0) whose path enclosed the minimal absolute area A1. For this purpose we fitted a parabola to the function A1(P(sl0)) using Brent’s method [14]. According to the golden section search the procedure jumped back and forth and found very fast the minimum value of A1(P(sl0)). This special procedure was considered to be very favourable because the searched point Pmin was commonly found to be closely positioned to point Pmin(sl0) (see below). Step 3: In order to find finally the point Pmin, we searched in the neighbourhood of the point Pmin(sl0) using Powell’s method [14] in two preferred directions: Vertical to the straight line sl0 and parallel to it. This procedure was a combination of multidimensional and one-dimensional minimization. Mostly it was sufficient to calculate the minimal value of the area A1 once vertical to the straight line sl0 and once again parallel. Rarely a second iteration was necessary.

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3. Results 3.1. The straight line sl0 In step 1 of our procedure, the function A2(x, y) was favourably fitted by the plane A2Plane(x, y) since A2(x, y) mostly represented nearly a plane. Figure 4 demonstrates this fact by the girl’s data (figure 1). The functions A2(x, y = +5 cm, 0, and –5 cm) nearly formed straight lines. Consequently the contour lines of A2(x, y) = constant nearly formed straight lines, too, which were parallel to the straight line sl0(A2 = 0) (figure 5).

Fig. 4. Data of patient KS: Areas A2 calculated from the paths of the net points P(x, y = const.) with y ∈ (+5 cm, 0, –5 cm). The graphs A2(x, y = const.) almost formed straight lines

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Fig. 5. Data of patient KS: A square of 10 10 cm2 was laid around the point PC. For net points of this square (distance = 0.1 cm) the paths and the corresponding areas A2 were calculated. In excellent approximation, the contour line graph (upper part) yielded straight lines including straight line sl0 (for area A2 = 0). A2(x, y) practically represented a plane (lower part)

For most of the 37 young and adult class-I-patients and the 28 young class-IIpatients the contour lines for A2 in the region of 10 10 cm2 were straight lines in the demonstrated good approximation. Hence, the approximation of the parting line l0 for the positive and negative areas A2 by the straight line sl0 was found to be the useful step 1 in our mathematical procedure. 3.2. The area A1min of the point Pmin The girl’s data yielded the absolute areas A1(sl0) for the points of the straight line sl0 (figure 6). Two minima were seen. The main minimum belonged to the point Pmin in figure 1; the second corresponded to the point P5.

Fig. 6. Data of patient KS: The absolute areas A1 were calculated for the paths of those points which lay on the straight line sl0. The curve showed two minima. The main minimum corresponded to the point Pmin, the second minimum belonged to the point P5 in figure 1

The absolute areas A1(x, y) were plotted for the net points in the 10 10 cm2 square (figure 7) as was already done in figure 5 for A2(x, y). The A1(x, y)-diagrams revealed a valley where the deepest point was the searched minimum. In the contour line graph, the straight line sl0 is added which was often found to run through the two minima as demonstrated in figure 6.

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Fig. 7. Data of patient KS: A square of 10 10 cm2 was laid around the point PC. For net points of this square (distance = 0.1 cm) the paths and the corresponding areas A1 were calculated. In the contour line graph (upper plot), a main minimum and a second minimum were found through which the straight line sl0 ran. The three-dimensional graph correspondingly showed two hollows in the valley (lower part)

Investigating the kinematical properties of the points Pmin and P5 in more detail we found that: 1. In all subjects with sound TMJs, Pmin moved exclusively forth in the course of mouth opening and exclusively back in the course of mouth closing. 2. Pmin normally ran apparently along the same path there and back. 3. The path of Pmin normally showed a circle-like curvature. These findings apparently checked with the expected properties of a MFHA. P5 (the second minimum) did not show these features: 1. The path was not at all circle-like. 2. The point already ran back during anterior mouth opening (figure 1). P5 was always caudal of Pmin and mostly found within the 10 10 cm2 square. In almost all cases, A1(P5) was larger than A1(Pmin). Since we had to reckon with two minima, we had to prevent the mathematical procedure of step 2 from taking the second as the searched minimum of A1(x, y). Brent’s method therefore started at first at the lower end of the straight line sl0 and then again at the upper end. The minimum positioned more cranially was always taken for the further evaluation. In step 3, we searched the absolute minimum of A1(x, y) in the region of the point Pmin(sl0) based on the findings that there the lines A1(x, y) = constant were almost parallel and therefore grad(A1(x, y)) was almost vertical to the straight line sl0 resulting from A2(x, y) = 0. Table. The statistical values for (a) the distance dmin between the point Pmin(sl0) and the point Pmin(A1min) and (b) the distance dref between the point PC and the point Pmin(A1min)

20

Device

K. M. THIEME et al. (a) Distance dmin between Pmin(sl0) and Pmin(A1min)

(b) Distance dref between PC and Pmin(A1min)

MT1602

Median cm 0.04

Mean cm 0.07

Std. dev. cm 0.07

Min cm 0.00

Max cm 0.36

Median cm 1.32

Mean cm 1.53

Std. dev. cm 0.81

Min cm 0.32

Max cm 3.66

CMS-JMA

0.02

0.04

0.05 F = 1.96 p < 5%

0.00

0.29

0.61

0.56

0.26 F = 9.71 p < 5%

0.04

1.11

The distance dmin between Pmin(A1min) and Pmin(sl0) was found to be small in each case (table), while the distance d ref between Pmin(A1min) and PC was large. Hence Pmin(A1min) was not located in the condyle’s centre. 3.3. Is the area A1min(Pmin) = 0? This question should be answered with yes if Pmin(A1min) was a point of an actually existing MFHA. Serious problems arose: On the one hand, the area A1min(Pmin) was unavoidably larger than zero because of measuring errors. On the other hand, we regarded the MFHA primarily not given by anatomical mechanical constraints but produced by piloting the mandible by the neuromuscular system. Because of this the condyles can be withdrawn a little bit from the os temporale giving thus the TMJ a certain articulating space [15]. Therefore we expected that the neuromuscular system was only able to implement a MFHA with certain uncertainty. To handle this problem we estimated the very upper limit of the error for the area A1min(Pmin)n of each person n and compared the distributions of these limits with that of the devices. The limits were determined in the following way: 1. We took the confidence interval (CI ) into account for determining a point: CI was 0.01 cm for the MT1602 [12] and 0.003 cm for the CMS-JMA [16]. 2. We calculated the individual path length Lpn by adding up the distances between the vertices CP(xi, yi) of the polygon (used for the calculation of A1min(Pmin)n) on the way there and back: L pn =

i =n−2 i =0

( xi +1 − xi ) 2 + ( yi +1 − yi ) 2 .

3. The rectangle Alimit n = CI · Lpn/2 was regarded as the individual upper limit of error. 4. The difference Dn = A1min(Pmin)n – Alimit n = A1min(Pmin)n – (CI · Lpn/2) was used as the test parameter to check whether the area A1min(Pmin) exceeded its upper limit of error.

Is a “movable hinge axis” used by the human stomatognathic system? a)

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b)

Fig. 8. Histograms of the A1min-distributions: (a) of the 37 young and adult class-I-patients measured with the MT1602 and of the 4 movement cycles of the young class-I-patient HK and (b) of the 28 young class-II-patients measured with the CMS-JMA and of the 8 movement cycles of the young class-II-patient KS. The upper limit of error was smaller than the median for the ensemble as well as for the individual

For both samples (MT1602 and CMS-JMA measurements) the median of the A1min(Pmin)-distribution lay above the respective very upper limit of error (figure 8a, b). In particular, for the more precise CMS-JMA measurements all individual A1min(Pmin)n lay above the upper limit of error. The Student-test of the variable Dn yielded: t = 8.25 (t(5%) = 1.99) for the MT1602 measurements and t = 12.58 (t(5%) = 2.00) for the CMS-JMA. Hence, the area A1min(Pmin) was found not to equal zero! This finding was supported by individual repeated measurements: t = 4.68 (t(5%) = 2.37) for patient HK and t = 8.40 (t(5%) = 2.13) for patient KS.

4. Discussion

All subjects made quasi-plane mandibular motions. The 3 DOF of horizontal and frontal rotation and lateral shift were measurable but small. We wondered whether the stomatognathic system used a further reduction from 3 to 2 DOF in a similar approximation. We could affirm this. The paths of the MFHA in mouth opening and closure almost coincided though the mandible was guided along strongly differing positions. The MT1602 and the CMS-JMA groups showed mean

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residual areas of ~0.048 cm² and ~0.025 cm², respectively, and path lengths of ~3.75 cm and ~2.09 cm. Hence the spacing between back and forth (0.048 cm2 / 3.75 cm = 0.013 cm and 0.025 cm2 / 2.09 cm = 0.012 cm, respectively) corresponded to the breadth of a thin pencil line. Therefore the neuromuscular systems were able to differentiate the entire ensemble of open mouth positions with high precision despite using practically 2 DOF. The mandibular positions could be definitely specified by two variables, the arc length L covered by the MFHA and the rotational angle in relation to the maxilla (figure 2). Thus an orthogonal coordinate system is obtained in which the ensemble of the possible positions of the rigid body is defined one-to-one. This system is independent of the coordinate system of the measuring apparatus and inherent in the stomatognathic system. It makes intra- and interindividual comparisons of mandibular movements possible and easy. Especially the structure of the guidance by the stomatognathic neuromuscular system can be evaluated when the mandible was brought from a starting to a final position [17]. Since according to Steiner the ensemble of the functions A2(x, y) = constant have to represent concentric circles and since we found in the most cases that A2(x, y) = 0 could be replaced by a straight line (sl0), the Steiner centre lay far away from the mandible. In the few cases having a nearer Steiner centre, the search for Pmin(A1min) required additional computational steps. The existence of A2(x, y) = 0 is common to plane kinematics with 3 DOF. It does not imply that a mandibular point Pmin must exist whose absolute area A1(Pmin) equals zero. A1(Pmin) = 0 is the criterion of plane movements with 2 DOF. The residual A1(Pmin) characterizes the approximation of the measured movements to plane movements with 2 DOF. It was found to be close. The data of the few patients with lacking precision and poor reproducibility in using the MFHA and having high residuals A1min(Pmin) gave hints that these patients have hidden problems with their neuromuscular apparatus. In this regard our method to evaluate in vivo the patients’ MFHA yields a novel diagnostic tool of clinical value by determining the area A1min(Pmin).

5. Conclusions Though an ideal MFHA could not be found, we could show that the stomatognathic system normally adjusts and uses a MFHA with a surprisingly high precision to guide the mandible’s position in the case of mouth opening and closing by using only 2 main DOF. Thus an inherent coordinate system of mandibular movements can be determined. The mathematical procedure for finding out the inherent coordinate system can be used for quasi-plane movements of other joint systems by analogy.

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Acknowledgements The authors wish to thank Professor Rainer Schwestka-Polly who did all the measurements with the MT1602. This work was supported by a grant from the Deutsche Forschungsgemeinschaft (SCHW 427/21, 2-2, and KU 535/6-3, respectively) and is permitted by the ethical commission (8/10/02).

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[8] NÄGERL H., KUBEIN-MEESENBURG D., FANGHÄNEL J., THIEME K. M., KLAMT B., SCHWESTKA-POLLY R., Elements of a general theory of joints. 6. General kinematical structure of mandibular movements, Anatomischer Anzeiger, 1991, Vol. 173, pp. 249–264.